SOLUTION: We have a random variable X = total number of success in n trials. If X is b(n,p) and assume n is some fixed number and p is to be estimated. Define the random variable p hat = X/

Question 1174870: We have a random variable X = total number of success in n trials.
If X is b(n,p) and assume n is some fixed number and p is to be estimated. Define the random variable p hat = X/n. Show that P hat is an unbiased estimator of p.
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Let's break down how to show that p̂ (p-hat) = X/n is an unbiased estimator of p.
**1. Definitions**
* **X:** The total number of successes in n trials.
* **X ~ b(n, p):** X follows a binomial distribution with n trials and probability of success p.
* **p̂ = X/n:** The estimator for p.
* **Unbiased Estimator:** An estimator is unbiased if its expected value is equal to the true parameter it's estimating. In this case, we need to show E[p̂] = p.
**2. Expected Value of X**
* Since X ~ b(n, p), the expected value of X is:
* E[X] = np
**3. Expected Value of p̂**
* We want to find E[p̂] = E[X/n].
* Using the linearity of expectation, we can write:
* E[p̂] = E[X/n] = (1/n) * E[X]
**4. Substitute E[X]**
* Substitute E[X] = np into the equation:
* E[p̂] = (1/n) * (np)
**5. Simplify**
* E[p̂] = (np)/n
* E[p̂] = p
**Conclusion**
Since E[p̂] = p, we have shown that p̂ = X/n is an unbiased estimator of p. This means that, on average, the sample proportion (p̂) will equal the true population proportion (p).

RELATED QUESTIONS
Suppose we have a binomial distribution with n trials and probability of success p. The... (answered by )

Assume the random variable X has a binomial distribution with the given probability of... (answered by ikleyn)

Assume the random variable X has a binomial distribution with the given probability of... (answered by MathLover1,math_tutor2020)

Assume the random variable X has a binomial distribution with the given probability of... (answered by Theo)

Assume the random variable X has a binomial distribution with the given probability of... (answered by ikleyn,ewatrrr)

Assume the random variable X has a binomial distribution with the given probability of... (answered by MathLover1)

Determine the binomial probability formula given the number of trials and the success... (answered by stanbon)

What is the variance of a binomial distribution in which the number of trials n = 100 and (answered by jim_thompson5910)